Logically impossible: Fatal condition
Preface
This Blogger article was originally published 23122017, slight revisions 20251214.
Fatal condition
The continuation of text review:
Key symbols
≡df = Equivalence by definition
: = Equal (s)
ε = Epsilon and means is
⊃ = Is the same as
⊨ is Entails
˜ = Not
∃ = There exists
∃! = There exists
∴ = Therefore
· = Therefore
< = Is included
v = a logical inclusive disjunction (disjunction is the relationship between two distinct alternatives).
x = variable
· = Conjunction meaning And
0 = Null class cls
= Class int
= Interpretation
---
Langer explains that a proposition can only be known via another proposition. (183). Implication is a relation that only holds among propositions. (183). Propositions are regarded as postulates. (185).
BLACKBURN, SIMON (1996) Oxford Dictionary of Philosophy, Oxford, Oxford University Press. LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)
The continuation of text review:
Key symbols
≡df = Equivalence by definition
: = Equal (s)
ε = Epsilon and means is
⊃ = Is the same as
⊨ is Entails
˜ = Not
∃ = There exists
∃! = There exists
∴ = Therefore
· = Therefore
< = Is included
v = a logical inclusive disjunction (disjunction is the relationship between two distinct alternatives).
x = variable
· = Conjunction meaning And
0 = Null class cls
= Class int
= Interpretation
---
Langer explains that a proposition can only be known via another proposition. (183). Implication is a relation that only holds among propositions. (183). Propositions are regarded as postulates. (185).
2025 note: But a proposition can simply be a statement on its own. This is a proposition that is not part of an argument. An argument requires at least one proposition, known as a premise, in the context of an argument, and a conclusion. For my Wales, MPhil and PhD degrees, I wrote propositions connected to questionnaires and surveys.
End
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A postulate needs to belong to the system, in the language of that system.
A postulate should imply further propositions of that system.
A postulate should not contradict any other accepted postulate, or any other proposition implied by another postulate. (185).
In other words, symbolic logic requires non-contradiction within its system in a universe of discourse.
Requirements
Coherence: Every proposition in the system must cohere to the established conceptual structure. (185). It must be in coherence with the rest.
Contributiveness : A postulate should contribute and have implication. (185-186).
Consistency: Most important states Langer (186). Two contradictory propositions (or premises) cannot contradict each other in a system. (186). The inconsistent is logically impossible. It is a fatal condition. (186). It is not logic at all. (186).
Independence: Postulates should be independent from each other. (186). If a proposition is deductible from a postulate already provided, then it is a theorem, a necessary fact, not another assumption. (186). Something provable in a theorem would be error to include as a postulate. (186).
I would reason that within philosophy there would be plenty of debate on what is a proposition/premise within systems and what would be a theorem. Langer explains that when a theorem needs elucidation, any proposition implied by another proposition as granted and proved within a system is a theorem. (186-187).
Within a biblical, system and universe of discourse...
Gd = God
(∃! Gd)
God exists. Would be viewed as necessary and a theorem.
Within an atheistic, system and universe of discourse...
˜ (∃! Gd)
God does not exist. Would be viewed as necessary and a theorem.
Noted: Some atheists would state they do not know if God exists, and not definitely that God does not exist. But my example stands as valid. Some have beliefs which actually are within a range of being atheists/agnostics and do fit within my proposition.
A postulate needs to belong to the system, in the language of that system.
A postulate should imply further propositions of that system.
A postulate should not contradict any other accepted postulate, or any other proposition implied by another postulate. (185).
In other words, symbolic logic requires non-contradiction within its system in a universe of discourse.
Requirements
Coherence: Every proposition in the system must cohere to the established conceptual structure. (185). It must be in coherence with the rest.
Contributiveness : A postulate should contribute and have implication. (185-186).
Consistency: Most important states Langer (186). Two contradictory propositions (or premises) cannot contradict each other in a system. (186). The inconsistent is logically impossible. It is a fatal condition. (186). It is not logic at all. (186).
Independence: Postulates should be independent from each other. (186). If a proposition is deductible from a postulate already provided, then it is a theorem, a necessary fact, not another assumption. (186). Something provable in a theorem would be error to include as a postulate. (186).
I would reason that within philosophy there would be plenty of debate on what is a proposition/premise within systems and what would be a theorem. Langer explains that when a theorem needs elucidation, any proposition implied by another proposition as granted and proved within a system is a theorem. (186-187).
Within a biblical, system and universe of discourse...
Gd = God
(∃! Gd)
God exists. Would be viewed as necessary and a theorem.
Within an atheistic, system and universe of discourse...
˜ (∃! Gd)
God does not exist. Would be viewed as necessary and a theorem.
Noted: Some atheists would state they do not know if God exists, and not definitely that God does not exist. But my example stands as valid. Some have beliefs which actually are within a range of being atheists/agnostics and do fit within my proposition.
However:
Ag = Agnostics
(∃! Ag)
Agnostics exist. Would be viewed as necessary and a theorem.
Cited
'Agnostic atheism – or atheistic agnosticism – is a philosophical position that encompasses both atheism and agnosticism. Agnostic atheists are atheistic because they do not hold a belief in the existence of any deity, and they are agnostic because they claim that such existence of a divine entity or entities is either unknowable in principle or currently unknown in fact.'
Key sources
Harrison, Alexander James (1894). The Ascent of Faith: or, the Grounds of Certainty in Science and Religion. London: Hodder and Stroughton. p. 21. OCLC 7234849. OL 21834002M.
Smith, George H (1979). Atheism: The Case Against God. Prometheus Books. pp. 10–11. ISBN 9780879751241.
Barker, Dan (2008). Godless: How an Evangelical Preacher Became One of America's Leading Atheists. New York: Ulysses Press. p. 96. ISBN 9781569756775. OL 24313839M.
End citations
Further
RS = Revelatory Scripture
(∃! RS) ⊨ (∃! Gd)
Revelatory Scripture exists entails God exists. From a Christian worldview, revealed, supernaturally inspired Scripture entails that God exists. Could another supernatural source provide supernatural revelation? Yes. God is reasoned to be God based on divine claims and supernatural interactions within that revelatory scripture that is considered religious history.
RS = Revelatory Scripture
(∃! RS) ⊨ (∃! Gd)
Revelatory Scripture exists entails God exists. From a Christian worldview, revealed, supernaturally inspired Scripture entails that God exists. Could another supernatural source provide supernatural revelation? Yes. God is reasoned to be God based on divine claims and supernatural interactions within that revelatory scripture that is considered religious history.
CONWAY DAVID A. AND RONALD MUNSON (1997) The Elements of Reasoning, Wadsworth Publishing Company, New York.
LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)
PIRIE, MADSEN (2006)(2015) How To Win Every Argument, Bloomsbury, London.
